Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics‐constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning and various industrial domains, especially in computational science and engineering. In the present survey, several promising avenues of research will be examined which focus on the combination of machine learning (ML) and domain decomposition methods (DDMs). The aim of this survey is to provide an overview of existing work within this field and to structure it into domain decomposition for machine learning and machine learning-enhanced domain decomposition, including: domain decomposition for classical machine learning, domain decomposition to accelerate the training of physics-aware neural networks, machine learning to enhance the convergence properties or computational efficiency of DDMs, and machine learning as a discretization method in a DDM for the solution of PDEs. In each of these fields, we summarize existing work and key advances within a common framework and, finally, discuss ongoing challenges and opportunities for future research.

Deep neural networks (DNNs) and, in particular, convolutional neural networks (CNNs) have brought significant advances in a wide range of modern computer application problems. However, the increasing availability of large numbers of datasets and the increasing available computational power of modern computers have led to steady growth in the complexity and size of DNN and CNN models, respectively, and thus, to longer training times. Hence, various methods and attempts have been developed to accelerate and parallelize the training of complex network architectures. In this work, a novel CNN-DNN architecture is proposed that naturally supports a model parallel training strategy and that is loosely inspired by two-level domain decomposition methods (DDMs). First, local CNN models, that is, subnetworks, are defined that operate on overlapping or nonoverlapping parts of the input data, for example, subimages. The subnetworks can be trained completely in parallel and independently of each other. Each subnetwork then outputs a local decision for the given machine learning problem which is exclusively based on the respective local input data. Subsequently, in a second step, an additional DNN model is trained which evaluates the local decisions of the local subnetworks and generates a final, global decision. With respect to the analogy to DDMs, the DNN models can be loosely interpreted as a coarse problem and hence, the new approach can be interpreted as a two-level domain decomposition. In this paper, we apply the proposed architecture to image classification problems using CNNs. Experimental results for different two-dimensional image classification problems are provided, as well as a face recognition problem and a classification problem for three-dimensional computed tomography (CT) scans. Therefore, classical Residual Network (ResNet) and VGG architectures are considered. More modern architectures, such as, e.g., MobileNet2, are left for future work. The results show that the proposed approach can significantly accelerate the required training time compared to the global model and, additionally, can also help to improve the accuracy of the underlying classification problem.

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.